REPRESENTATION TYPE OF QUANTUM SCHUR ALGEBRAS

KARIN ERDMANN AND DANIEL K. NAKANO

Abstract. In this paper we classify the q-Schur algebras having finite, tame or wild representation type and also the ones which are semisimple.

1 Introduction

1.1 Let A be a finite-dimensional algebra over a field k. Furthermore, let mod(A) denote the category of finite-dimensional A-modules. The algebra A has finite representation type if and only if there are only finitely many indecomposable modules in mod(A). Otherwise, A has infinite representation type. Algebras of infinite representation type can be divided into two mutually exclusive categories-ones which are either have tame representation type or wild representation type. An algebra A has tame representation type if for each dimension d there are finitely many 1-parameter families of modules which contains all d-dimensional indecomposable modules. In this situation there is a good chance one may be able to describe all indecomposable objects in mod(A). On the other hand, algebras of wild representation type are those algebras whose representation theory is as complicated as modules over the free associative algebra k⟨x, y⟩. Given a class of algebras a fundamental problem is to classify the representation type of the algebras and to gain as much insight into those of finite or tame type as possible.

Let GL_{n}(k) be the general linear group over k and V be the n-dimensional natural representation. The Schur algebra S(n, d) is defined as End_{Σ_{d}}(V^{⊗d}) where Σ_{d} is the symmetric group on d letters. These algebras arise naturally in the study of the polynomial representation theory of GL_{n}(k). In [Erd2], [DN], and [DEMN], a complete classification of the representation type for the Schur algebra was obtained. The purpose of this paper is to provide a complete classification of the representation type of the quantum (or q) Schur algebras.

1.2 Definitions and Notation: Let R be an integral domain and q be a unit in R, we assume q ≠ 1. Let us regard the symmetric group Σ_{r} as the set of bijections of {1, …, r}. The Hecke algebra H_{q}(r) = H_{R,q}(Σ_{r}) is the free R-module with basis {T_{w} : w ∈ Σ_{r}}. The multiplication in the algebra is defined by the rule

T_{w}T_{s} = \begin{cases}T_{ws} & if ℓ(ws) > ℓ(w) \\ qT_{ws} + (q − 1)T_{w} & otherwise \end{cases}

Date: September 2000.

Research of the second author partially supported by NSF grant DMS-9800960.

where s = (i, i + 1) ∈ Σ_{r} is a basic transpositions and w ∈ Σ_{r}. The function ℓ : Σ_{r} → ℕ is the usual length function.

For each partition λ of r there is a q-Specht module of the Hecke algebra H_{q}(r), denoted by S^{λ}. Assume that R = k is a field and that q ∈ k with q ≠ 0, 1.

Let λ be a composition of r. and let Σ^{λ} be the corresponding Young subgroup of Σ_{d}, so Σ^{λ} ≅ Σ_{λ_{1}} × Σ_{λ_{2}} × · · ·. Let

x_{λ} = ∑_{w∈Σ^{λ}} T_{w}.

Define M^{λ} := x_{λ}H_{q}(r). The module M^{λ} has a unique submodule isomorphic to the Specht module S^{λ}, and there is a unique indecomposable direct summand of M^{λ} containing S^{λ}; this is the Young module Y^{λ}. We have Y^{λ} ≅ Y^{μ} if and only if λ = μ. Each M^{α} (for α ∈ Λ(n, r) ) is a direct sum of such Young modules (see [Mar]).

The q-Schur algebra was introduced by Dipper and James as follows. Let Λ(n, r) be the set of compositions of r into n parts. The q-Schur algebra is defined as the algebra

S_{q}(n, r) = S_{R,q}(n, r) = End_{H_{q}(r)} ( ⨁_{λ∈Λ(n,r)} x_{λ}H_{q}(r)) .

Note that if q = 1 with R = k where k is a field, then this is the classical Schur algebra S(n, r) = S_{k}(n, r) [Gr].

The q-Schur algebra over a field can also be constructed via quantum GL_{n}; here we use the version introduced in [DD]; and we take advantage of the work done in [Don]. Let A_{q}(n) be the bialgebra defined there; for each r ≥ 0 its homogeneous part A_{q}(n, r) is a finite-dimensional subcoalgebra, and its dual is therefore a finite-dimensional algebra which can be identified with the q-Schur algebra. We note that the representation theory works well for arbitrary non-zero q.

1.3 Main results: For the remainder of the paper we will assume that R = k is an algebraically closed field of characteristic p ≥ 0 (p = 0 or a prime number). We make the convention that p is arbitrary unless it is explicitly specified. We also assume q ∈ k and q ≠ 0, 1; and n ≥ 2. Let l be the multiplicative order of q in k^{∗}, that is either q is a primitive l-th root of 1, or l = ∞. Note that if q is a primitive l-th root of 1 and p > 0 then l and p must be coprime. The main results of the paper are listed below.

Theorem (A). The algebra S_{q}(n, r) is semisimple if and only if one of the following holds:

(i) q is not a root of unity; (ii) q is a primitive lth root of unity and r < l; (iii) n = 2, p = 0, q is a primitive 2nd root of unity, and r = 2w + 1 with w ∈ ℕ; (iv) n = 2, p ≥ 3, q is a primitive 2nd root of unity, and r = 2w + 1 with w < p.

Theorem (B). The algebra S_{q}(n, r) has finite representation type but is not semi-simple if and only if q is a primitive lth root of unity with l ≤ r, and one of the following holds:

(i) n ≥ 3 and r < 2l; (ii) n = 2, p ≠ 0, l ≥ 3 and r < lp; (iii) n = 2, p = 0 and either l ≥ 3, or l = 2 and r is even; (iv) n = 2, p ≥ 3, l = 2 and r even with r < 2p, or r = 2w + 1 (odd) with p ≤ w < p^{2}.

Theorem (C). The algebra S_{q}(n, r) has tame representation type if and only if q is a primitive lth root of unity and one of the following holds:

(i) n = 3, l = 3, p ≠ 2 and r = 7, 8; (ii) n = 3, l = 2 and r = 4, 5; (iii) n = 4, l = 2 and r = 5; (iv) n = 2, l ≥ 3, p = 2 or p = 3 and pl ≤ r < (p + 1)l;

(v) n = 2, l = 2, p = 3 and r ∈ {6, 19, 21, 23};

We remark that Theorem 1.3(A) corrects several misconceptions in the literature on the semisimplicity of quantum Schur algebras (see [PW, (11.4.1) Lem.] [Mar, p. 191 Rem. (1)] [Don, p. 96 (7)]). It is also interesting to note that when p = 0 the algebra S_{q}(2, r) is always semisimple, or of finite representation type.

1.4 For the sake of convenience it will be useful to reformulate the results given in the preceding section.

Theorem (A). The algebra S_{q}(n, r) is not semi-simple if and only if q is a primitive l-th root of unity, such that r ≥ l and one of the following holds:

(a) n ≥ 3;

(b) n = 2 and l ≥ 3;

(c) n = 2, l = 2 and r even;

(d) n = 2, l = 2, p ≥ 3 and r = 2w + 1 with w ≥ p.

Theorem (B). The algebra S_{q}(n, r) is of infinite type if and only if q is a primitive l-th root of unity and one of the following holds:

(a) n ≥ 3 and r ≥ 2l;

(b) n = 2, p ≠ 0, l ≥ 3 and r ≥ lp;

(c) n = 2, p ≥ 3, l = 2 and r is even, r ≥ 2p, or r is odd, r ≥ 2p^{2} + 1.

Theorem (C). The algebra S_{q}(n, r) has wild representation type if and only if q is a primitive l-th root of unity and one of the following holds:

(a) n ≥ 3, l ≥ 4 and r ≥ 2l;

(b) n ≥ 3, l = 3 and r ≥ 2l except for the algebras in Theorem 1.3(C)(i);

(c) n ≥ 3, l = 2 and r ≥ 2l, except for the algebras in Theorem 1.3(C)(ii) and (iii);

(d) n = 2, l ≥ 3, p ≠ 0 and r ≥ lp except for the algebras in Theorem 1.3(C)(iv);

(e) n = 2, l = 2, p ≥ 3 and r is odd, r ≥ 2p^{2} + 1, or r is even and r ≥ 2p, except for the algebras in Theorem 1.3(C)(v).

We start with general reductions (in §2). In §3, we analyze the case when n = 2. From these results we can present a complete proof of Theorems 1.3(A) and 1.4(A). In §4, we provide the proofs Theorems 1.3(B), 1.3(C), 1.4(B) and 1.4(C), modulo studying the representation type of specific q-Schur algebras S_{q}(n, d) for n = 2, 3, 4. The remaining verifications will be accomplished in §5-6.

1.5 Acknowledgements: The second author would like to acknowledge the Mathematical Institute at Oxford University for their support and hospitality during his visit in June 1997. We also thank the referee for many valuable suggestions.

2 Preliminaries

2.1 Let A be a finite dimensional algebra. We say that A has property (*) if A has one of the following properties:

• A is not semisimple

• A has infinite representation type

• A has wild representation type

2.2 Let A and B be two finite-dimensional algebras. Let F be an additive functor from mod(A) to mod(B) which preserves indecomposables and reflects isomorphisms. If such a functor exists then it is easy to see that (i) if A is not semisimple then B is not semisimple, provided F is also either left exact or right exact, (ii) if A has infinite representation type then B has infinite representation type, and (iii) if A is wild then so is B (this follows directly from the characterization via a bimodule). In many situations such a functor can be found.

2.3 The following two propositions give sufficient conditions for recognizing an algebra of wild representation type.

Proposition (A). Let A be a finite-dimensional algebra with primitive orthogonal idempotents e_{1} and e_{2}. Let P = e_{1}A and e be the idempotent e = e_{1} +e_{2}. Suppose eP has radical quotient rad_{eAe}(eP)/ rad_{eAe}^{2}(eP) = S_{1} ⊕ S_{1} ⊕ S_{2} where S_{i} is the simple corresponding to e_{i}. Then A is wild.

Proof. It suffices to show that eAe/rad_{eAe}^{2}(eAe) is wild. Consider the composition of the following functors: first, the natural functor from mod(eAe/rad^{2}(eAe)) to mod(A), then the functor − ⊗_{eAe} eA. Clearly the first functor has the properties as in 2.2, and the second one as well which can be seen by considering its right adjoint. By the hypothesis, the quiver for this algebra contains a connected quiver with two vertices and two loops at one of the vertices (see Figure 1). Any such algebra is wild by comparison with the connected quiver with three vertices and three arrows whose dual is listed in [Ri1].

[figure: •; S_{1}; •; S_{2}] Figure 1
□

Proposition (B). Let A be a finite-dimensional algebra such that Ext_{A}^{1}(S, T) ≅ Ext_{A}^{1}(T, S) for any two simple modules S, T of A. Suppose the quiver of A contains a proper subquiver which is a circuit with at least 3 vertices. Then A is wild.

Proof. The separated quiver for A contains Ã_{n} which arises from this circuit. This is a proper subquiver and the condition on the extensions implies that the separated quiver contains Ã_{n} extended by one arrow. Therefore, A is wild. One can see this by exhibiting a functor from the category of representations of E^{≈}_{7} into the category of A-modules which preserves indecomposability and reflects isomorphisms. _{□}

2.4 We will now focus our attention on q-Schur algebras. The results in this subsection will be used to reduce the general question of representation type of q-Schur algebras to looking at the representation type of S_{q}(n, r) for n = 2, 3, 4, 5 and small r. Our first result compares the representation type of S_{q}(n, r) with the representation type of S_{q}(n′, r) where n′ ≥ n.

Proposition (A). If S_{q}(n, r) has property (*) then S_{q}(n′, r) has property (*) for n′ ≥ n.

Proof. There exists an idempotent e in S_{q}(n′, r) such that eS_{q}(n′, r)e ≅ S_{q}(n, r) [Don, 2.2(1) p.48]. Set A = S_{q}(n′, r) and eAe = S_{q}(n, r). Let F be the functor defined from mod(eAe) to mod(A) by F (−) = Ae ⊗_{eAe} (−). This functor preserves indecomposables and reflects isomorphisms, and is right exact. Furthermore, from [Erd1, I 4.7], if eAe has wild representation type, then so does A. _{□}

Let M be a (right) S_{q}(n, r)-module, N a k⟨x, y⟩-module, and Q an S(n, r′) − k⟨x, y⟩- bimodule which is free as right k⟨x, y⟩-module. With these assumptions M ⊗_{k} Q becomes an S_{q}(n, r + r′) − k⟨x, y⟩-bimodule. The left action on this tensor product is given by the bialgebra structure on quantum matrix space. On the other hand the right actions of k⟨x, y⟩ is given by (m ⊗ q).a = m ⊗ (q.a) for m ∈ M, q ∈ Q and a ∈ k⟨x, y⟩. Moreover, there exists an isomorphism of S_{q}(n, r_{1} + r_{2})-modules:

M ⊗_{k} [Q ⊗_{k⟨x,y⟩} N] ≅ [M ⊗_{k} Q] ⊗_{k⟨x,y⟩} N.  (2.4.1)

Let δ be the one-dimensional determinant representation of quantum GL_{n} which also may be regarded as a representation of S(n, n). We will use this representation to relate the representation type of S_{q}(n, r) with S_{q}(n, r + n).

Proposition (B). If S_{q}(n, r) has property (*) then S_{q}(n, n + r) has property (*).

Proof. The statement of the proposition concerning semisimplicity and infinite representation type will hold if there exists an exact functor from mod(S_{q}(n, r)) to mod(S_{q}(n, n + r)) which preserves indecomposability and reflects isomorphisms. The functor defined by G(−) = δ ⊗ (−) has the required properties.

Now suppose that S_{q}(n, r) has wild representation type. Then there exists a S_{q}(n, r) − k⟨x, y⟩-bimodule Q such that the functor F (−) = Q ⊗_{k⟨x,y⟩} − preserves indecomposability and reflects isomorphisms. Consider the functor F′ from mod(k⟨x, y⟩) to mod(S_{q}(n, r + n)) defined by F′(−) = [δ ⊗_{k} Q] ⊗_{k⟨x,y⟩} − .

We will first show that F′ preserves indecomposability. Assume that N is an indecomposable k⟨x, y⟩-module and F′(N) ≅ X_{1} ⊕ X_{2}. Working in the category of rational representations, we have

(δ^{−1} ⊗ X_{1}) ⊕ (δ^{−1} ⊗ X_{2}) ≅ δ^{−1} ⊗_{k} F′(N)
≅ δ^{−1} ⊗_{k} [[δ ⊗_{k} Q] ⊗_{k⟨x,y⟩} N]
≅ Q ⊗_{k⟨x,y⟩} N
≅ F (N)

Therefore, since F preserves indecomposability F′ must also preserve indecomposability.

Next we will prove that F′ reflects isomorphisms. Let F′(N_{1}) ≅ F′(N_{2}). By tensoring both sides of this isomorphism by δ^{−1} we have Q ⊗_{k⟨x,y⟩} N_{1} ≅ Q ⊗_{k⟨x,y⟩} N_{2}. The functor F preserves isomorphisms so N_{1} ≅ N_{2}. Hence, S_{q}(n, r + n) has wild representation type. _{□}

Let GL_{n,q} denote quantum GL_{n}. Moreover, let X(T) be the weight lattice and X(T)_{+} be the set of dominant integral weights. For any GL_{n,q}-module V and λ, μ ∈ X(T), define pr_{λ}V to be the largest GL_{n,q}-submodule of V such that μ ∈ W_{p} · λ for every composition factor L_{q}(μ) of V. Let Φ^{+} be a set of positive roots for the root system of type A_{n−1}, ρ be the half-sum of positive roots and

C̄_{ℤ} = {λ ∈ X(T) | 0 ≤ ⟨λ + ρ, α^{∨}⟩ ≤ p for all α ∈ Φ^{+}}.

Moreover, let λ, μ ∈ C̄_{ℤ} and let ν be the unique weight in X(T)_{+}∩W (μ−λ). The translation functor T_{λ}^{μ} : mod(GL_{n,q}) → mod(GL_{n,q}) is defined by

T_{λ}^{μ}(V) = pr_{μ}(L_{q}(ν) ⊗ pr_{λ}V).

Let P_{q}(n) be the category of polynomial modules of GL_{n,q} and let F be the functor from mod(GL_{n,q}) to P_{q}(n) defined by taking the largest P_{q}(n)-submodule of a given GL_{n,q}- module. For any λ, ν ∈ C̄_{ℤ}, define a truncated translation functor T̂_{λ}^{μ} to be the composite F ∘ T_{λ}^{μ}. By restriction T̂_{λ}^{μ} induces a functor from P_{q}(n) into itself.

Proposition (C). Let B(λ) be a block of S_{q}(n, r) containing the simple module L_{q}(λ). Suppose that μ ∈ Λ^{+}(n, r′) be a weight in the same facet as λ with μ − λ ∈ P (D). If B(λ) has property (*) then S_{q}(n, r′) has property (*).

For the definition of “facet”, we refer to [Jan, p.261], and P (D) is the set of polynomial weights, which we identify with ℕ^{n}.

Proof. Under the hypotheses given in the theorem T̂_{μ}^{λ} ∘T̂_{λ}^{μ} is equivalent to the identity functor. This shows that the functor T̂_{λ}^{μ} preserves indecomposable modules and isomorphism classes. Therefore, the statement of this proposition concerning semisimplicity and infinite representation type holds.

Suppose that B(λ) has wild representation type. Let F be the functor from mod(k⟨x, y⟩) to mod(B(λ)) defined by F (−) = Q ⊗_{k⟨x,y⟩} − where Q is a B(λ) − k⟨x, y⟩-bimodule which is free as right k⟨x, y⟩-module. Furthermore, let F′ be the functor from mod(k⟨x, y⟩) to mod(B(μ)) defined by F′(−) = T̂_{λ}^{μ}(Q)⊗_{k⟨x,y⟩}−. We have T̂_{λ}^{μ}(Q) is a B(μ)−k⟨x, y⟩-bimodule which is also free as k⟨x, y⟩-module, also a direct verification shows that F′ = T̂_{λ}^{μ} ∘ F.

We will first show that F′ takes indecomposables to indecomposables. Let N be an indecomposable k⟨x, y⟩-module. Let F′(N) = T̂_{λ}^{μ} ∘F (N) ≅ X_{1} ⊕X_{2} where X_{j} ≠ {0} for j = 1, 2. Since T̂_{μ}^{λ} ∘ T̂_{λ}^{μ} is equivalent to the identity functor, we have T̂_{μ}^{λ}(X_{1}) ⊕ T̂_{μ}^{λ}(X_{2}) ≅ F (N). Observe that T̂_{μ}^{λ}(X_{j}) ≠ {0} for j = 1, 2. Since X_{j} ≠ 0, there exist simple modules L_{q}(γ_{j}) with γ_{j} ∈ W_{p} · μ which are submodules of X_{j} for j = 1, 2. Therefore, 0 ≠ T̂_{μ}^{λ}(L_{q}(γ_{j})) ↪ T̂_{μ}^{λ}(X_{j}) for j = 1, 2. This contradicts the fact that F preserves indecomposability, thus F′ must preserve indecomposability.

Finally, we show that F′ preserves isomorphism classes. Assume that F′(N_{1}) ≅ F′(N_{2}). Therefore,

F (N_{1}) ≅ T̂_{μ}^{λ}(T̂_{λ}^{μ}F (N_{1})) ≅ T̂_{μ}^{λ}(T̂_{λ}^{μ}F (N_{2})) ≅ F (N_{2})

and N_{1} ≅ N_{2}, and F′ preserves isomorphism classes. Hence, B(μ) must have wild representation type. _{□}

2.5 The q-Schur algebras are quasi-hereditary algebras [PW, 11.5] [Mar, Thm. 7.5.1]. In examining the representation theory of the q-Schur algebras we will extensively make use of the representation theory of quasi-hereditary algebras [Ri2] [Don, Appendix]. A quick review of this subject is presented below.

Let A be a finite-dimensional algebra A with simple modules L(λ) labelled by λ ∈ Λ where (Λ, ≤) is a partially ordered set. Let P (λ) be the projective cover of L(λ). The standard module Δ(λ) is defined to be the largest quotient of the projective P (λ) which has only composition factors L(μ) with μ ≤ λ. The costandard module ∇(λ) is defined similarly, as a submodule of the injective hull I(λ) of L(λ). The algebra A is quasi-hereditary if for each λ ∈ Λ:

(2.5.1) L(λ) occurs only once as a composition factor of Δ(λ);

(2.5.2) P (λ) has a filtration with sections isomorphic to standard modules where Δ(λ) occurs only once, and if Δ(μ) occurs then μ ≥ λ.

Quasi-hereditary algebras satisfy “Brauer-Humphreys reciprocity”. Let [P (λ) : Δ(μ)] denote the number of times Δ(μ) appears as a section in a Δ-filtration of P (λ).

(2.5.3) Then [P (λ) : Δ(μ)] = [Δ(μ) : L(λ)].

Let F(Δ) be the category of all A-modules which have a Δ-filtration; and similarly F(∇) be the category of all A-modules which have a ∇-filtration. Then for each λ ∈ Λ there is a unique indecomposable module denoted by T (λ) with highest weight λ which has both a Δ-filtration and also a ∇-filtration [Ri2] [Don]. Moreover these are all indecomposable modules which belong to F(Δ) ∩ F(∇). We call any module which is a direct sum of such T (λ) a tilting module.

Let T := ⊕_{λ}T (λ)^{n(λ)} with n(λ) ≥ 1 for all λ. Consider the algebra A′ := End_{A}(T), which is again quasi-hereditary. This algebra is known as the Ringel dual of A, and is unique up to Morita equivalence. Furthermore, (A′)′ is Morita equivalent to A. According to [Ri2], the functor F = Hom_{A}(T, −) induces an equivalence F_{A}(∇) → F_{A′}(Δ), taking ∇(λ) to Δ′(λ) and T (λ) to the projective P′(λ). It follows that (2.5.4) [T (λ) : ∇(μ)] = [P′(λ) : Δ′(μ)] = [Δ′(μ) : L′(λ)]

where the last step uses the reciprocity law for A′ . Since (A′)′ is Morita equivalent to A with standard modules identified, it also follows that

(2.5.5) [T′(λ) : ∇′(μ)] = [P (λ) : Δ(μ)] = [Δ(μ) : L(λ)].

If A is quasi-hereditary then Ext_{A}^{1}(Δ(γ), Δ(μ)) ≠ 0 implies γ < μ. Moreover for μ ≱ λ

Hom_{A}(rad_{A}Δ(λ), L(μ)) ≅ Ext_{A}^{1}(L(λ), L(μ)).

In particular, Ext_{A}^{1}(L(λ), L(λ)) = 0.

Suppose that A has a duality (−)^{0} fixing the simple modules. For the q-Schur algebra this is given by the transposed dual. In this case Δ(λ) ≅ (∇(λ))^{0} and also T (λ)^{0} ≅ T (λ). Therefore, [T (λ) : ∇(μ)] = [T (λ) : Δ(μ)]. Furthermore, if A is a quasi-hereditary algebra with duality then for any λ, μ we have Ext_{A}^{1}(L(λ), L(μ)) ≅ Ext_{A}^{1}(L(μ), L(λ)). Since λ ≠ μ implies either μ ≱ λ or λ ≱ μ we deduce that the quiver of A can be obtained by knowing the second radical layer of the standard modules.

By considering the Ringel duality from A′ to A one sees that if T_{A′}(λ) is injective then the A-module P (λ) must be a tilting module.

2.6 The reduction theorems in Section 2.4 will allow us to reduce questions of representation type to S_{q}(n, r) for n = 2, 3, 4, 5 and r small. With these reductions we will make explicit computations of the quiver and relations for these algebras. This will be accomplished in the following way. In order to study the quiver and relations for the algebra S_{q}(n, r), it is enough to consider its basic algebra, that is the algebra

End_{S_{q}(n,r)}(⊕_{λ∈Λ^{+}(n,r)}P (λ))

where the set indexing the simples is Λ^{+}(n, r). Because we are looking at small rank cases it will be possible to determine the structure of the projective modules P (λ) for q-Schur algebras. ¿From this information one can then compute out the quiver and relations for this algebra. This technique was used effectively in [DEMN] for this purpose. In some instances we will refer to that paper for the detailed computations.

The standard modules are the usual q-Weyl modules which we denote by Δ(λ). There is a duality fixing the simple modules; hence the quiver of a q-Schur algebra can be obtained from knowing the structures of the q-Weyl modules. Often it will be enough for us to find the quiver. The q-Weyl modules we study will mostly be duals of q-symmetric powers. The submodule structure of these is completely understood by [Th, 3.4, 3.9]; we will use this frequently without further comments. In general, the dimension of Δ(λ) is given by Weyl’s character formula.

Another method for computing the quiver and relations for q-Schur algebras is to compute homomorphisms between Young modules. From Section 1.2 the algebra End(⊕_{λ∈Λ^{+}(n,r)}Y^{λ}) is a basic algebra for S_{q}(n, r). If B is a block of the Hecke algebra H_{q}(r) then let S_{B} be the endomorphism ring of ⊕_{D^{λ}∈B}Y^{λ}. The algebra S_{B} is the basic algebra for a sum of blocks of S_{q}(n, r) (usually just one block). As in [Erd2], for certain parameters r and B one determines the submodule structure of the Y^{λ} and by applying results from [Erd2] a presentation for S_{B} by quivers and relations is obtained. We recall that the blocks of the Hecke algebra are described by the ’Nakayama conjecture’ (see [JM]).

The structure of the Y^{λ} is to some extent also found via the q-Schur algebra (the method in [Erd2] was different). The Young modules have a Specht filtration, and to understand their structure we need to know firstly the filtration multiplicities and secondly the decomposition numbers [S^{μ} : D^{λ}] for the Hecke algebra.

The Young module Y^{λ} is projective if and only if λ is l-restricted. However the reciprocity formula [Y^{λ} : S^{μ}] = [Δ(μ) : L(λ)] holds for arbitrary λ: There is a quotient of the Hecke algebra for which the modules S^{λ} and Y^{λ} for 2-part partitions λ survive, which is isomorphic to a Ringel dual of S_{q}(2, r) (almost, if l = 2) in such a way that the Young modules are identified with the tilting modules and the Specht modules with the standard modules. (cf. [Don, 4.7]). Therefore this identity holds by (2.5.4). Similarly, if λ is l-regular then

[S^{μ} : D^{λ}] = [T (λ) : Δ(μ)].

The filtration multiplicities of T (λ) can be studied from its structure as twisted tensor products; and for n = 2 this gives a complete answer ([Don, 3.3, 3.4], see also §3.1). As a further tool we use row-and column removal, from ([Don, 4.2(9), (15)]).

If Y^{λ} is projective and if all partitions in the block of λ are l-regular the projective P (λ) for the Schur algebra is a tilting module.

2.7 In [DEMN, 3.5], a basic algebra was defined by quiver and relations, which will be denoted by call S. The quiver for S is given in Figure 2, with the relations α_{1}β_{1} = α_{2}β_{2} = β_{3}α_{3} = 0, α_{1}α_{2}α_{3} = 0 = β_{3}β_{2}β_{1}, β_{1}α_{1}α_{2} = α_{2}α_{3}β_{3}, and β_{2}β_{1}α_{1} = α_{3}β_{3}β_{2}. When the characteristic of the field is 2 then the classical Schur algebra S(2, 6) is Morita equivalent to S, and it was shown to have wild representation type. This algebra will arise in several different places so it would be nice to be able to identify S by simply knowing the decomposition matrix and some conditions on the projectives.

[figure: •; L(4); •; L(3); •; L(1); •; L(2); α_{1}; β_{1}; α_{2}; β_{2}; α_{3}; β_{3}] Figure 2

Proposition . Let A be a quasi-hereditary algebra with duality fixing the simple modules. Moreover, assume that A has four simple modules L(i) where i ∈ {1, 2, 3, 4} and < is the natural order. Suppose A has decomposition matrix of the form

(\begin{matrix}[Δ(4) : L(μ)] \\ [Δ(3) : L(μ)] \\ [Δ(2) : L(μ)] \\ [Δ(1) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix})

where the rows and columns are indexed by decreasing weights. Also, assume that P (1) and P (2) are injective. Then A is Morita equivalent to S. Furthermore, S has wild representation type.

Proof. Injective modules for quasi-hereditary algebras have ∇-filtrations. Therefore, if P (1) is injective then it is a tilting module and self-dual, so it is isomorphic to I(1). By reciprocity, the filtration sections of P (1) can be seen from the last column of the matrix. Using Extproperties for quasi-hereditary algebras, it follows that P (1) has a filtration

P (1) = M_{0} ⊃ M_{1} ⋯ ⊃ M_{4} = 0

such that M_{i−1}/M_{i} ≅ Δ(i). So L(4) is the composition factor with the highest weight in P (1) hence P (1) ≅ T (4). Moreover we have that Δ(4) is a submodule of P (1) ≅ I(1), so the socle of Δ(4) is simple and isomorphic to L(1). It follows that Δ(4) is uniserial, with heart L(3).

We claim that M_{2} ≅ P (3). The module M_{2} is indecomposable since its socle is simple, and it has the same Δ-quotients as P (3), so it suffices to show that Ext_{A}^{1}(Δ(3), Δ(4)) ≅ k. This is clear once one applies Hom_{A}(−, Δ(4)) to the exact sequence

0 → Δ(4) → P (3) → Δ(3) → 0  (2.7.1)

In particular it follows that P (3) has a simple socle L(1).

Since P (2) is also injective, it follows that Δ(3) has socle L(2) and is therefore uniserial, and P (2) must be uniserial with composition factors

L(2), L(1), L(3), L(1), L(2)

With this information it is straightforward to verify the relations. The algebra S is wild because P (1) has a quotient T (1) which is uniserial with composition factors L(1), L(2), L(1) and it also has a uniserial quotient with composition factors L(1), L(3), L(1). It follows that End_{A}(P (1)) is a Kronecker algebra. Now apply the Proposition 2.3(A) with e_{1} and e_{4}. _{□}

3 Representation type for S_{q}(2, r)

3.1 We will first review some facts on q-Weyl modules and tilting modules for quantum GL_{2}, from [Don] and [Cox2]. Let q be an l-th root of unity where l > 1 and let k be a field of characteristic p or of characteristic zero. Moreover, let F : GL_{n,q} → GL_{n} be the Frobenius map. The following notation will be used for weights [Cox2, 3.3] when n = 2. For a restricted partition λ = (λ_{1}, λ_{2}) we set δ = λ_{2} and μ = λ_{1} − λ_{2}. Thus

λ = (μ + δ, δ)  (3.1.1)

where 0 ≤ μ ≤ l − 1 and δ ≥ 0. We consider now weights satisfying 0 ≤ μ ≤ l − 2. Define μ̄ by μ + μ̄ = l − 2. Next we define

λ̃ := (λ_{2} − 1, λ_{1} + 1 − l) = μ̄ρ + (μ − l + 1 + δ)ϖ  (3.1.2)

where ρ = (1, 0) and ϖ = (1, 1).

First consider the Weyl modules. We have for n ≥ 1

Δ(ln − 1 + δ, δ) ≅ Δ(l − 1 + δ, δ) ⊗ Δ((n − 1)ρ)^{F} .  (3.1.3)

Moreover there is a non-split exact sequence of G-modules

0 → Δ(n − 1)ρ)^{F} ⊗ Δ(λ̃ + lϖ) → Δ(λ + lnρ) → Δ(nρ)^{F} ⊗ Δ(λ) → 0 (3.1.4)

Since here Weyl modules for restricted highest weights are simple, (3.1.3) and (3.1.4) determine completely the composition factors for Weyl modules.

Now let us consider the tilting modules. If λ is restricted then Δ(λ) is simple and hence isomorphic to T (λ). The tilting module T (λ̃ + lρ) has two Δ-quotients, namely Δ(λ̃ + lρ) and Δ(λ). If τ = (λ̃ + lρ) + lα (where α is an arbitrary partition) then we have

T (τ) ≅ T (λ̃ + lρ) ⊗ T (α)^{F}  (3.1.5)

This allows one to work inductively since T (λ̃ + lρ) ⊗ Δ(nρ)^{F} has a Δ-filtration; there is an exact sequence

0 → Δ(λ̃ + (n + 1)lρ) → T (λ̃ + lρ) ⊗ Δ(nρ)^{F} → Δ(λ + lnρ) → 0.  (3.1.6)

3.2 Let S = S_{q}(2, r) where q is a primitive l-th root of unity. Every block component S_{B} is Morita equivalent to the block component of a principal block of some q-Schur algebra S_{q}(n, d) for some d ≤ r. To see this, apply the method in [Erd2, 1.3] (for the q-analogue, see [Don, 4.2(18)]). If B is not principal then this induces an isomorphism of S_{B} with S_{q}(2, r − 2)B̃ where B̃ is the block whose core is obtained from that of B by removing the first column. If the new core has only one part then B̃ is principal; otherwise repeat the process.

Let r = wl + l − 1. Then the functor X → X^{F} ⊗ St from the category of modules for the classical Schur algebra S(n, w) to the category of S_{q}(n, r)-modules gives an equivalence of S(n, w)−mod with the principal block component of S_{q}(n, r), this is the q-analogue of [Jan, II.10.5]; a variation of [Cox1, 5.4].

Lemma . The principal block component of S_{q}(2, wl + l − 1) is Morita equivalent to the classical Schur algebra S(2, w), over the same field.

3.3 We consider the principal block component of S_{q}(2, r) for small r in detail. Let r = wl + t where w ≥ 0 and 0 ≤ t ≤ l − 1. Let B be the block of H_{q}(r) with core (t). In the case when t < l − 1, the component S_{B} contains w + 1 simple modules, labelled by the following 2-part partitions.

λ(s) = (r − sl, sl) for 0 ≤ s ≤ w/2 or (w − 1)/2  (3.3.1)
μ(s) := (r − sl − t − 1, sl + t + 1) for 0 ≤ s ≤ w/2 − 1 or (w − 1)/2. (3.3.2)

Denote by J_{m} the m × m Jordan block matrix which is upper triangular matrix with ones along the diagonal and superdiagonal.

Proposition (A). Let r = wl + t where 0 ≤ t ≤ l − 1, and 0 < w. If k has characteristic p then also assume that w ≤ p − 1.

(a) If t = l − 1 then S_{B} is semisimple.

(b) Assume t ≠ l − 1. Then S_{B} is a block of S_{q}(2, r) which is Morita equivalent to the algebra A_{w+1}.

We recall from [Erd2] the definition of A_{m}. It has quiver given in Figure 3 and is defined by the following relations:

α_{i}α_{i+1} = 0, β_{i+1}β_{i} = 0, α_{1}β_{1} = 0
β_{i}α_{i} = α_{i+1}β_{i+1}, (1 ≤ i ≤ m − 2).

[figure: •; L(1); •; L(2); •; L(3); •; L(m − 1); •; L(m); α_{1}; β_{1}; α_{2}; β_{2}; …; α_{m−1}; β_{m−1}] Figure 3

Proof. Part (a) holds by Lemma 3.2 since w ≤ p − 1 so that S(2, w) is semisimple.

(b) Assume r = wl + t where 0 ≤ t < l − 1. We consider first the decomposition matrix for the Specht modules of the Hecke algebra H_{q}(r) of the partitions listed in (3.3.1), (3.3.2) above. This has the form (J_{w+1})^{t}, for w ≤ p, with the exception that if l = 2 and the degree is even then the last column has to be removed. This can be proved by using the tilting modules (see §2.6, §3.1 and [Don, 4.4(6)]).

The filtration multiplicities of the Young modules are equal to decomposition numbers for Weyl modules, see §2.6. We find that for w < p, this matrix has form (J_{w+1})^{t}. For w = p it is almost (J_{w+1})^{t} except that the first row is [1, 1, 1, 0, ⋯ 0]. This comes from the fact that the module Δ(lp) has length three; and the GL_{2}(k)-module Δ(p) is not simple anymore.

Note that Young modules are self-dual. Therefore it follows easily from the above mulitplicities, that the Young modules for the Hecke algebra have the structure of the modules in [Erd2, (3.2)]. The statement of the result now follows from [Erd2, (3.2)]. _{□}

Proposition (B). Assume that k has characteristic p ≥ 2. Let r = pl + t where t = 0 or 1, where r ≢ −1(mod l). Then the block component S_{B} is Morita equivalent to the algebra D_{p+1}.

Proof. The submodule structure of almost all Young modules can be obtained as in the proof of the previous proposition. The structures coincide with those in [Erd2, 3.6] except for the one labelled by (pl − l, l) has three Specht quotients, namely (pl, 0) and (pl − 1, 1) and (pl − l, l). By the same argument as in [Erd2] we see that it has the structure as the module denoted by E_{3} in [Erd2, (3.6)] and the results follows. _{□}

Proposition (C). Let p = 2 and let S = S_{q}(2, r) and S_{B} be the component corresponding to the principal block.

(a) If r = 2l + t where 0 ≤ t < l − 1, then S_{B} is Morita equivalent to D_{3}.

(b) If r = 2l + l − 1, then S_{q}(2, r) has a block which is Morita equivalent to D_{3}.

(c) If r = 3l + t where t = 0, 1 then S_{B} is Morita equivalent to the algebra S (see §2.7).

Proof. (a) This is a special case of Proposition 3.3(B).

(b) Take the block component which corresponds to core (l−2, 1). This is Morita equivalent to the principal block component of S_{q}(2, r − 2) as in §3.2.

(c) We consider the case when r = 3l; the other is similar. ¿From 3.1 we get that the decomposition matrix for the principal block is of the form

(\begin{matrix}[Δ(3l) : L(μ)] \\ [Δ(3l − 1, 1) : L(μ)] \\ [Δ(2l, l) : L(μ)] \\ [Δ(2l − 1, l + 1) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix})

The tilting modules T (3l) and T (3l − 1, 1) are both projective and injective and are isomorphic to to P (2l − 1, l + 1) and P (2l, l) respectively (see [CE, 6.4]). The results follows by Proposition 2.7. _{□}

Proposition (D). Let p = 3 and let S_{B} be the component of S := S_{q}(2, r) corresponding to the principal block.

(a) If r = 3l + t where 0 ≤ t < l − 1 then S_{B} is Morita equivalent to D_{4}.

(b) If r = 3l + l − 1 and l ≠ 2, then S has a block which is Morita equivalent to D_{4}.

(c) If r = 4l, or if r = 4l + 1 and l > 2, then S_{B} is wild.

Proof. (a) The case t = 0, 1 was done in Proposition 3.3(B), the other t are similar. For (b), take the block with core (l − 2, 1), see §3.2.

(c) First we consider the case when r = 4l. The decomposition matrix for the principal block component is given by

(\begin{matrix}[Δ(4l) : L(μ)] \\ [Δ(4l − 1, 1) : L(μ)] \\ [Δ(3l, l) : L(μ)] \\ [Δ(3l − 1, l + 1) : L(μ)] \\ [Δ(2l, 2l) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

In fact, from [Th, 3.4, 3.9], Δ(4l − 1, 1) is uniserial with composition factors L(4l − 1, 1), L(3l − 1, l + 1), L(3l, l), and Δ(4l) has a simple socle isomorphic to L(3l − 1, l + 1) and (of course) a simple top, isomorphic to L(4), and radΔ(4l)/socΔ(4l) is the direct sum of L(2l, 2l) and L(4l − 1, 1). This shows that the quiver has a “circuit” of length four and an extra edge. Therefore, the algebra is wild by Proposition 2.3(B).

The case r = 4l + 1 but l ≠ 2 uses the same argument; the decomposition matrix where the columns and rows are indexed by the partitions (4l+1), (4l−1, 2), (3l+1, l), (3l−2, l+2) and (2l + 1, 2l) have the same entries as in the r = 4l case. _{□}

3.4 Proof of Theorem 1.3(A): We can now present a complete proof of Theorem 1.3(A). Recall that we assume q ≠ 1. Also, if q is a primitive l-th root of unity and the characteristic of the field is p > 0 then l and p are coprime.

We will show that the algebras listed in Theorem 1.3(A) are semi-simple, and to complete the proof we will show that the algebras listed in Theorem 1.4(A) are not semi-simple.

(3.4.1) [1.3(i) and 1.3(ii)] If q is not a root of unity, or q is an l-th root of unity and r < l then S_{q}(n, r) is semi-simple:

The Hecke algebra is semisimple when q is not a root of unity or when q is a primitive lth root of unity and r < l by a result of Dipper and James [DJ2]. Therefore, S_{q}(n, r) is semisimple because the endomorphism algebra of a completely reducible module is semisimple.

From now we assume that q is a primitive l-th root of 1.

(3.4.2) [1.3(iii) and 1.4(d)] Let l = 2 and r = 2w + 1. If p = 0 then S_{q}(2, r) is semi-simple. Moreover, if p ≥ 3 then S_{q}(2, r) is semisimple if and only if w < p.

Let p = 0, the first part of the statement holds by Proposition 3.3(A)(a) and results in §3.2. Let p ≥ 3. Then S_{q}(2, r) has two block components, by §3.1. We know that the prinicpal block component is Morita equivalent to the classical Schur algebra S(2, w) and this is semisimple if and only if w < p (see [DN]). So for w ≥ p the algebra S_{q}(2, r) is not semisimple. Now let w < p; the other block component of S_{q}(2, r) is Morita equivalent to the principal block component of S_{q}(2, r − 2), hence Morita equivalent to S(2, w − 1), and we get that for w < p the algebra S_{q}(2, r) is semisimple.

Now assume r ≥ l.

(3.4.3) [1.4(b) and most of 1.4(a)] If l ≥ 3 and n ≥ 2 then S_{q}(n, r) is not semi-simple for r ≥ l.

Consider S_{q}(2, l) and S_{q}(2, l + 1) for l ≥ 3. These algebras are not semisimple by Proposition 3.3(A)(b). Therefore, S_{q}(2, r) is not semisimple for r ≥ l by Proposition 2.4(B), and S_{q}(n, r) is not semisimple for l ≥ 3, n ≥ 2, and r ≥ l by Proposition 2.4(A).

(3.4.4) [1.4(a)] If l = 2 and n ≥ 3 then S_{q}(n, r) not semisimple for r ≥ l

By Proposition 3.3(A)(b) we know that S_{q}(2, 2) and S_{q}(2, 4) are not semisimple, hence S_{q}(3, 2) and S_{q}(3, 4) are not semisimple. Moreover, S_{q}(3, 3) is not semisimple because Δ(3) is not simple (but indecomposable). Hence, S_{q}(n, r) is not semisimple for n ≥ 3 and r ≥ l by Propositions 2.4(A) and 2.4(B).

(3.4.5) [1.4(c)] If l = 2 then S_{q}(2, r) is not semisimple for r = 2w ≥ 2 where w ∈ ℕ.

Consider S_{q}(2, 2) for l = 2. This algebra is not semisimple by Proposition 3.3(A)(b). Therefore, S_{q}(2, r) is not semisimple for r = 2w ≥ 2 where w ∈ ℕ by Proposition 2.4(B).

4 Representation type for S_{q}(n, r), n ≥ 3

4.1 Let l ≥ 3 and consider the principal block component of the algebra S_{q}(3, 2l). We first claim that some of the decomposition numbers for Weyl modules are given by the following matrix.

(\begin{matrix}[Δ(2l) : L(μ)] \\ [Δ(2l − 1, 1) : L(μ)] \\ [Δ(l, l) : L(μ)] \\ [Δ(2l − 2, 1^{2}) : L(μ)] \\ [Δ(l, l − 1, 1) : L(μ)] \\ [Δ((l − 1)^{2}, 2) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 1 & 0(1) & 0 & 0 & 1 \\ 0 & 1 & ∗ & ∗ & ∗ & ∗ \\ 0 & 0 & 1 & ∗ & ∗ & ∗ \\ 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix})

If p ≠ 2 then the Weyl module Δ(2l) has length three and has the stated composition factors. Moreover, this module is uniserial with second socle L(2l − 1, 1) for p ≠ 2. For p = 2 there is an extra composition factor L(l, l), and it occurs in the second radical layer. In both cases, Δ(2l) contains a uniserial submodule of length two with top L(2l − 1, 1) and socle L((l − 1)^{2}, 2). The last three rows follow easily by consideration of dimensions, and by using Steinberg’s tensor product theorem.

Lemma . Let l ≥ 3. Then the prinicpal block component of S_{q}(3, 2l) is wild.

Proof. To see this we investigate the quiver. Consider the Schur algebra S_{q}(3, l) over the same field. The principal block component has the Weyl module Δ(l − 1, 1), of length two, with socle L(l−2, 1^{2}). By tensoring with L(1)^{F} we get an indecomposable module of length two with socle L(2l − 2, 1^{2}) and head L(2l − 1, 1). Using this and the information from the last three rows of the decomposition matrix together with the above mentioned submodule of Δ(2l) we see that the quiver contains a circuit with four vertices. Since the quiver is connected with 6 vertices, it contains this as a proper subquiver and hence the algebra is wild, by Proposition 2.3(B). _{□}

Proposition . If l ≥ 4 then S_{q}(3, r) has wild representation type for r = 2l, 2l + 1 and 2l + 2.

Proof. We have seen that the block component of S_{q}(3, 2l) containing L(2l − 2, 1^{2}) has wild representation type. Consider the blocks of S_{q}(3, 2l + 1) (resp. S_{q}(3, 2l + 2)) which contain the simple modules L(2l−2, 2, 1) (resp. L(2l−2, 2^{2})). Set λ = (2l−2, 1^{2}), μ_{1} = (2l−2, 2, 1) and μ_{2} = (2l − 2, 2^{2}). The weights λ, μ_{1} and μ_{2} belong to the same facet for l ≥ 4 and λ−μ_{j} ∈ P (D) for j = 1, 2. Consequently, by Proposition 2.4(C), it follows that S_{q}(3, 2l+1) and S_{q}(3, 2l + 2) have wild representation type. _{□}

4.2 We now provide a list of algebras for n = 3, 4, 5 along with their representation type. This is verified on a case by case basis in Sections 5 and 6.

(4.2.1) S_{q}(3, 7), p ≠ 2, l = 3 (tame, §5.1)

(4.2.2) S_{q}(3, 8), p ≠ 2, l = 3 (tame, §5.1)

(4.2.3) S_{q}(3, 10), p ≥ 0, l = 3, (wild, §5.2)

(4.2.4) S_{q}(3, 11), p ≥ 0, l = 3, (wild, §5.2)

(4.2.5) S_{q}(3, 7), p = 2, l = 3 (wild, §5.3)

(4.2.6) S_{q}(3, 8), p = 2, l = 3 (wild, §5.3)

16 KARIN ERDMANN AND DANIEL K. NAKANO

(4.2.7) S_{q}(3, 4), p ≥ 0, l = 2, (tame, §5.4) (4.2.8) S_{q}(3, 5), p ≥ 0, l = 2, (tame, §5.5) (4.2.9) S_{q}(3, 6), p ≥ 0, l = 2, (wild, §5.6) (4.2.10) S_{q}(3, 7), p ≥ 0, l = 2, (wild, §5.7) (4.2.11) S_{q}(3, 8), p ≥ 0, l = 2, (wild, §5.8) (4.2.12) S_{q}(4, 7), p ≠ 2, l = 3, (wild, §6.1) (4.2.13) S_{q}(4, 8), p ≠ 2, l = 3, (wild, §6.2) (4.2.14) S_{q}(4, 4), p ≥ 0, l = 2, (wild, §6.3) (4.2.15) S_{q}(4, 5), p ≥ 0, l = 2, (tame, §6.4) (4.2.16) S_{q}(5, 5), p ≥ 0, l = 2, (wild, §6.5)

4.3 Proof of Theorem 1.3(B): Assume now that S_{q}(n.r) is not semi-simple. That is, q is a primitive l-th root of unity and r ≥ l. Moreover if n = 2 and l = 2 then either r is even, or p ≥ 3 and r ≥ 2p + 1. We will show that the algebras listed in Theorem 1.3(B) have finite type, and that the algebras listed in Theorem 1.4(B) are of infinite type. Note that in the following the semisimple algebras are excluded. (4.3.1) [1.3(i)] If r < 2l then S_{q}(n, r) has finite representation type.

If r < 2l then all blocks of H_{q}(r) are Brauer tree algebras of finite type, or are semisimple ([U], see [EN]). The proof of [Erd2, (4.1)] works here as well and we deduce that S_{q}(n, r) has finite type. (4.3.2) [1.3(ii) and part of 1.3(iv)] If p ≠ 0 then S_{q}(2, r) has finite representation type for r < lp.

Let l ≥ 2. As explained in §3.2, every block component of S_{q}(2, r) is Morita equivalent to a principal block component of some S_{q}(2, d) for some d ≤ r [Don]. Let d = wl + t ≤ r where 0 ≤ w and 0 ≤ t ≤ l − 1, so that w < p. By Proposition 3.3(A) we have that the principal block component of S_{q}(2, d) is semi-simple or of finite type. It follows that S_{q}(2, r) has finite type for r < lp. (4.3.3) [1.3(iii)] If p = 0 then S_{q}(2, r) has finite representation type (if it is not semi-simple). This follows from Proposition 3.3(A) and the results in §3.2. (4.3.4) [rest of 1.3(iv) and part of 1.4(c)] If p ≠ 0 and l = 2 then S_{q}(2, r) has infinite representation type for r ≥ 2p^{2} + 1 and has finite representation type for r = 2w + 1 where p ≤ w < p^{2}.

Let r = 2w + 1 and p ≤ w. For l = 2, we have from §3.2 that the principal block component of S_{q}(2, r) is Morita equivalent to the classical Schur algebra S(2, w), over characteristic p. The other block component is Morita equivalent to the principal block component of S_{q}(2, r − 2). Therefore, by [Erd2] S_{q}(2, 2w + 1) has finite representation type if w ∈ {1, 2, … p^{2} − 1}, and infinite representation type if w ≥ p^{2}. Now apply Proposition 2.3(B).

(4.3.5) [1.4(a)] If n ≥ 3 and r ≥ 2l then S_{q}(n, r) has infinite representation type.

By Proposition 2.4(A) and Proposition 2.4(B) it suffices to prove this for n = 3 and r = 2l, 2l + 1 and r = 2l + 2. This holds by Proposition 4.1 for l ≥ 4, and by Lemma 4.1, (4.2.1), (4.2.2), (4.2.5), and (4.2.6) for l = 3, and by (4.2.7), (4.2.8) and (4.2.9) when l = 2.

(4.3.6) [1.4(b)] If l ≥ 3 and p ≠ 0 then S_{q}(2, r) has infinite representation type for r ≥ lp.

Let p ≥ 2. By Proposition 3.3(B) the algebra S_{q}(2, lp) has infinite type, for l ≥ 2. If l ≥ 3 then also S_{q}(2, lp + 1) has infinite type, by Proposition 2.4(C) and hence for l ≥ 3 and any r ≥ lp the algebra S_{q}(2, r) has infinite type.

(4.3.7) [rest of 1.4(c)] If p ≥ 3, l = 2 and r is even then S_{q}(2, r) has infinite representation type for r ≥ 2p.

The algebra S_{q}(2, 2p) has infinite representation type by Proposition 3.3(B); now apply Proposition 2.3(B).

4.4 Proof of Theorem 1.3(C): We assume that S_{q}(n, r) is of infinite type, so that one of the conditions in Theorem 1.4(B) holds. We show that the algebras listed in Theorem 1.3(C) are tame, and moreover that the algebras in Theorem 1.4(C) are wild.

(4.4.1) [1.3(i-iii)] S_{q}(n, r) has tame representation type for n = 3, l = 3, p ≠ 2 and r = 7, 8; n = 3, l = 2, p ≥ 0 and r = 4, 5; n = 4, l = 2, p ≥ 0 and r = 5.

The tameness for S_{q}(n, r) for n = 3, l = 3, p ≠ 2 and r = 7, 8 holds by (4.2.1) and (4.2.2); for n = 3, l = 2, p ≥ 0 and r = 4, 5 by (4.2.7) and (4.2.8); for n = 4, l = 2, p ≥ 0 and r = 5 by (4.2.15).

(4.4.2) [1.3(iv)] S_{q}(2, r) has tame representation type for l ≥ 3, p = 2, 3 and pl ≤ r < (p+1)l.

We can apply Proposition 3.3(C) and Proposition 3.3(D), together with results §3.2. The algebras involved are Morita equivalent to either D_{3} or D_{4} which are tame by [DEMN, §5.1, §5.5].

(4.4.3) [1.3(v) and part of 1.4(e)] If p = 3, l = 2 (so that r ≥ 6) then S_{q}(2, r) is tame for r ∈ {6, 19, 21, 23} and wild for r even and r ≥ 8 and for r odd and r ≥ 25.

If r is even then S_{q}(2, r) is tame for r = 6, by Proposition 3.3(D) and it is wild for r = 8 and hence wild for all even r ≥ 8. Now let r = 2w + 1; by §3.2, the principal block component of S_{q}(2, 2w + 1) is Morita equivalent to the classical Schur algebra S(2, w) and the other block component is Morita equivalent to the classical Schur algebra S(2, w − 1). By [DEMN], over characteristic 3 the classical Schur algebra S(2, m) is tame if and only if m = 9, 10, 11, it is wild for m ≥ 12 and of finite type for m ≤ 8.

To complete the proof we must show that the remaining algebras are wild. Assume now that 2l ≤ r.

(4.4.4) [1.4(a)] Assume n ≥ 3 and l ≥ 4, then S_{q}(n, r) is wild.

By Proposition 2.3(B) and Proposition 2.3(A) it suffices to show this for n = 3 and r = 2l, 2l + 1, 2l + 2. This was verified in Proposition 4.1.

(4.4.5) [1.4(b)] Assume that n ≥ 3 and l = 3. Then S_{q}(n, r) is wild unless p ≠ 2 and r = 7, 8.

Consider first the case p ≠ 2. If r = 6 and n = 3 then by Lemma 4.1 the algebra is wild, and then also for r = 3k ≥ 6. If r = 10, 11 and n = 3 then by (4.2.3) and (4.2.4) the algebra is wild. Combining these it follows, using Propositions 2.4(A) and 2.4(B), that for all n ≥ 3 and r ≥ 9, the algebra S_{q}(n, r) is wild. If n = 4 and r = 7, 8 then by (4.2.12) and (4.2.13) the algebra is wild. Together with the previous we deduce that S_{q}(n, r) is wild for n ≥ 4 and r ≥ 6.

Now assume p = 2. Then S_{q}(3, r) is wild for r = 6, 7, 8 by Lemma 4.1, (4.2.5) and (4.2.6). Using Propositions 2.3(A) and 2.3(B) it follows that S_{q}(n, r) is wild for n ≥ 3 and r ≥ 6 when p = 2.

(4.4.6) [1.4(c)] Let n ≥ 3 and l = 2 (so that r ≥ 4). Then S_{q}(n, r) is wild unless n = 3 and r = 4, or 3 ≤ n ≤ 4 and r = 5.

If n = 4 and r = 4 then the algebra is wild by (4.2.14), and if n = 5, r = 5 it is wild by (4.2.16). Now suppose n = 3 and r = 6, 7, 8 then the algebra is wild by (4.2.9), (4.2.10), and (4.2.11). These imply 1.4(c), using Propositions 2.3(A) and 2.3(B).

(4.4.7) [1.4(d)] Let n = 2 and l ≥ 3 (so that r ≥ pl, p ≠ 0). Then S_{q}(2, r) is wild unless p ≤ 3 and pl ≤ r < (p + 1)l.

If p ≥ 5 then apply Proposition 3.3(B) which shows that S_{q}(2, r) is wild for r ≥ lp. For p = 2 and r = 3l, 3l + 1 use Proposition 3.3(C) and Proposition 2.7, so S_{q}(2, r) is wild for r ≥ 3l. Suppose p = 3, then use Proposition 3.3(D) and it follows that S_{q}(2, r) is wild for r ≥ 4l.

(4.4.8) [rest of 1.4(e)] Let l = 2 and n = 2, and p ≠ 0 and assume that either r is even ≥ 2p, or r is odd ≥ 2p^{2} + 1. Then S_{q}(2, r) is wild unless p = 3 and r ∈ {6, 19, 21, 23}.

By Proposition 3.3(B) the algebra is wild for p ≥ 5 and r = 4, 5, As explained in (4.4.3) it is wild for p = 3, r even ≥ 8 and for r odd, ≥ 25.

5 Representation type for some S_{q}(3, r)

5.1 In the next three sections we will deal with algebras S_{q}(3, r) where l = 3. First we consider the case when the characteristic of the field is not two. We recall the definition of the algebras R_{4} and H_{4}. The algebra R_{4} is given by the quiver in Figure 4 with relations

α_{1}β_{1} = 0, α_{1}α_{2} = 0 = β_{2}β_{1}, β_{1}α_{1} = α_{2}β_{2}, β_{2}α_{2} = α_{3}β_{3}.

The algebra H_{4} has quiver in Figure 5 and is defined by the following relations:

\begin{matrix}αδ = 0 = γβ, αβ = 0 = αρ, \\ ζβ = 0 = ζρ, δγ = βα + ρζ. \end{matrix}

[figure: •; •; •; •; α_{1}; β_{1}; α_{2}; β_{2}; α_{3}; β_{3}] Figure 4

[figure: •; X_{1}; •; X_{2}; •; X_{3}; •; X_{4}; α; β; δ; γ; ζ; ρ] Figure 5

Proposition . Let p ≠ 2 and l = 3. The basic algebra of the principal block of S_{q}(3, r) is Morita equivalent to R_{4} [Erd2, 3.11] if r = 7 and H_{4} [Erd2, 3.13] if r = 8. Furthermore, under these conditions the algebras S_{q}(3, 7) and S_{q}(3, 8) are tame.

Proof. First let r = 7 and consider the principal block component. The aim is to show that the hypotheses for [Erd2, 3.12] are satisfied for the Young modules. To get the filtration multiplicities for the Young modules by reciprocity (see §2.6), we start by finding the decomposition matrix for the Weyl modules. We claim that it is of the form

(\begin{matrix}[Δ(7) : L(μ)] \\ [Δ(5, 2) : L(μ)] \\ [Δ(4, 3) : L(μ)] \\ [Δ(4, 2, 1) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix})

The first row can be obtained from [Th]. By row or column removal one gets the third row and also [Δ(5, 2) : L(4, 3)] = 1. Then there is only one missing entry which follows by consideration of dimensions.

Next we will show that the decomposition matrix for 3-part partitions of H_{q}(7) when l = 3 (and p ≠ 2) is given as follows:

(\begin{matrix}[S^{(7)} : D^{μ}] \\ [S^{(5,2)} : D^{μ}] \\ [S^{(4,3)} : D^{μ}] \\ [S^{(4,2,1)} : D^{μ}] \end{matrix}) = (\begin{matrix}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{matrix})

First we know that Y^{(5,2)} has Specht quotients S^{(5,2)} and S^{(7)} which gives the second row. Next we have [S^{(4,3)} : D^{(5,2)}] = 1 by column removal and the decomposition matrix for H_{q}(3). We see that D^{(4,3)} is 1-dimensional and that by dimensions, the first entry of the row of [S^{(4,3)} : D^{μ}] must be 0.

By row removal we get [S^{(4,2,1)} : D^{(4,3)}] = 1. Moreover Y^{(4,2,1)} is projective and must be P (D^{(5,2)}) and hence by reciprocity [S^{(4,2,1)} : D^{(5,2)}] = 1. The Specht module S^{(4,1^{3})} is irreducible (see [Jam, 6.22] ), and is isomorphic to D^{α} for some α ⊳ (4, 1^{3}). Moreover, this module has dimension 20 and the only possibility is α = (4, 2, 1). This gives also the last entry, that is [S^{(4,2,1)} : D^{(7)}], by dimensions.

By reciprocity we see that Y^{(4,3)} has Specht quotients S^{(5,2)} and S^{(4,3)} and since it is selfdual it must have a simple top D^{(5,2)}. Hence Y^{(4,3)} is a quotient of P (5, 2) and the kernel is necessarily equal to S^{(4,2,1)} (one inclusion since D^{(4,2,1)} does not occur and equality by length). This gives the second radical layer of P (5, 2), which is k ⊕ D^{(4,3)}. It follows that S^{(4,2,1)} has “diamond shape” with socle D^{(5,2)}. The dimensions of Hom_{H_{q}(7)}(Y^{λ}, Y^{μ}) are characteristic-independent (see [DJ1]), and are seen by considering the Specht-quotients. We get the remaining hypothesis for [Erd2, 3.12] and deduce that the block component of S_{q}(3, 7) is Morita equivalent to R_{4}, as stated. There are two more blocks which are both of finite type (which is easy to see). Hence, S_{q}(3, 7) is tame.

Now consider the algebra S_{q}(3, 8) for l = 3 where p ≠ 2. The aim is to verify the hypotheses of [Erd2, 3.14] for the principal block component. The decomposition matrix for the Weyl modules is

(\begin{matrix}[Δ(8) : L(μ)] \\ [Δ(5, 3) : L(μ)] \\ [Δ(5, 2, 1) : L(μ)] \\ [Δ(4, 3, 1) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix})

The verification is similar to the S_{q}(3, 7) case. It follows that the Young modules corresponding to the partitions (8) and (5, 3) are simple and hence so are the corresponding Specht modules. The Young module Y^{(5,2,1)} has a “cross shape” (cf. [Erd2, 3.14]). This gives the decomposition numbers for the Specht module S^{(5,2,1)}. Moreover Y^{(4,3,1)} = P (D^{(5,3)}). The decomposition matrix for 3-part partitions of the Hecke algebra is therefore given by:

(\begin{matrix}[S^{(8)} : D^{μ}] \\ [S^{(5,3)} : D^{μ}] \\ [S^{(5,2,1)} : D^{μ}] \\ [S^{(4,3,1)} : D^{μ}] \end{matrix}) = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ ∗ & 1 & 1 & 1 \end{matrix})

Note that the multiplicity of D^{(5,2,1)} in S^{(4,3,1)} is seen by column removal. The Specht module S^{(4,3,1)} ⊂ P (D^{(5,3)}) must be contained the square of the radical. Then it follows that the remaining decomposition number is zero and that the submodule structure of P (D^{(5,3)}) is as needed for [Erd2, 3.14].

Therefore, one can conclude that that the principal block of S_{q}(3, 8) is Morita equivalent to the algebra H_{4} as in [Erd2]. This has tame representation type by [DEMN, 5.7]. For the representation type of the entire algebra one must also consider the other blocks. The block containing L(6, 1^{2}) is Morita equivalent to A_{2}. This leaves us with the block containing L(7, 1). By the same arguments as for S_{q}(3, 7) (in the first part of the proof) one shows that this block component is Morita equivalent to the algebra R_{4}, and it follows that S_{q}(3, 8) is tame for l = 3 and p ≠ 2. _{□}

5.2 We will now consider the algebras S_{q}(3, 10) and S_{q}(3, 11) for l = 3. The following two results show that these algebras of wild representation type independent of the characteristic of the field.

Proposition (A). The algebra S_{q}(3, 10) is wild when l = 3.

Proof. If p = 2 and l = 3 then by Proposition 3.3(C) S_{q}(2, 10) is wild, hence S_{q}(3, 10) is wild by Proposition 2.4(A). So assume that p ≠ 2. The decomposition matrix for the Weyl modules in the principal block component is given as follows:

(\begin{matrix}[Δ(10) : L(μ)] \\ [Δ(8, 2) : L(μ)] \\ [Δ(7, 3) : L(μ)] \\ [Δ(7, 2, 1) : L(μ)] \\ [Δ(5, 5) : L(μ)] \\ [Δ(4, 3^{2}) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix})

The tilting modules T (5, 5) and T (7, 2, 1) are both uniserial of length three, with top and socle L(4, 3, 3) and the middle is L(5, 5) or L(7, 2, 1) respectively. So they are both quotients of P (4, 3, 3). We deduce that there is an exact sequence

0 → W = Δ(7, 3) → P (4, 3^{2}) → X → 0

and an exact sequence

0 → Δ(5, 5) ⊕ Δ(7, 2, 1) → X → Δ(4, 3, 3) → 0

Let U be the quotient of P (4, 3^{2}) by rad(W). We claim that the socle of U is L(7, 3) ⊕ L(4, 3^{2}) ⊕ L(4, 3^{2}). This is clear from the above should it be the case that

Ext_{S_{q}(3,10)}^{1}(L(7, 3), L(4, 3^{2})) = 0.

Otherwise, L(7, 3) must occur in the top of the radical of P (4, 3^{2}) and then it also lies in the socle of U. This is enough to see that the algebra is wild. Namely, let e = e_{(4,3^{2})} + e_{(7,3)} where the e_{λ} are orthogonal primitive, with e_{λ}S_{q}(3, 10) = P (λ), and apply Proposition 2.3(A). _{□}

Proposition (B). The algebra S_{q}(3, 11) is wild when l = 3.

Proof. If p = 2 then S_{q}(2, 11) is wild, so S_{q}(3, 11) is wild. Let p ≠ 2. The decomposition matrix for the principal block component is

(\begin{matrix}[Δ(11) : L(μ)] \\ [Δ(8, 3) : L(μ)] \\ [Δ(8, 2, 1) : L(μ)] \\ [Δ(7, 3, 1) : L(μ)] \\ [Δ(5^{2}, 1) : L(μ)] \\ [Δ(5, 3^{2}) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & ∗ & ∗ & ∗ \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix})

The lower part of the matrix is obtained from the decomposition matrix of S_{q}(3, 8) over the same field by tensoring with the determinant.

The block of S_{q}(3, 8) containing L(7, 1) is known, it is Morita equivalent to the algebra R_{4}. This together with the structure of Δ(11) shows that the quiver contains a subquiver

L(11) ⟷ L(8, 2, 1) ⟷ L(7, 3, 1) ⟷ L(5^{2}, 1) ⟷ L(5, 3^{2}).

In fact, this is the full subquiver with these vertices. Assume (for a contradiction) that the algebra is not wild; then the vertex L(8, 3) cannot be joined to too many other simples, by 2.3(B). Since there exists a module with simple top and Δ-quotients Δ(8, 2, 1), Δ(8, 3) (the quotient P (8, 2, 1)/Δ(11)), it follows that L(8, 3) must be joined to L(8, 2, 1) or L(7, 3, 1) in the quiver; so that by 2.3(B) it is joined to precisely one of these and not to any other vertex of the quiver. This gives therefore the quiver completely.

Suppose the first possibility holds, then by considering dimensions and using the fact that Δ(8, 3) has a simple top, it follows that Δ(8, 3) can only be uniserial, and its composition factors must be

L(8, 3), L(8, 2, 1), L(7, 3, 1), L(5^{2}, 1)

This implies that P (8, 2, 1) has a uniserial quotient of length three with composition factors L(8, 2, 1), L(7, 3, 1), L(5^{2}, 1). All the weights μ occuring are ≤ (8, 2, 1), so this must be a quotient of the Weyl module Δ(8, 2, 1), a contradiction.

Now consider the second possibility. Recall from the structure of R_{4} that P (5, 3^{2}) has a quotient with radical layering

L(5, 3^{2})
L(5^{2}, 1)
L(7, 3, 1) ⊕ L(5, 3^{2})

Consider the representations of the wild quiver D^{≈}_{4} given in [Ri1]. There is a functor from the category of the representations of this quiver into the module category of the quiver given in Figure 6, and this can be identified with a subcategory of S_{q}(3, 11). The functor preserves indecomposables and reflects isomorphisms, hence S_{q}(3, 11) is wild. _{□}

[figure: L(11); L(8, 2, 1); L(7, 3, 1); L(5^{2}, 1); L(5, 3^{2}); L(8, 3); L(5, 3^{2})] Figure 6

5.3 Consider S_{q}(3, 7) and S_{q}(3, 8) for l = 3 and p = 2. We will show that both of these algebras are wild. First, we look at S_{q}(3, 7). The Weyl module Δ(7) is uniserial of length three, socle L(5, 2); and the decomposition matrix for the principal block component is given by

(\begin{matrix}[Δ(7) : L(μ)] \\ [Δ(5, 2) : L(μ)] \\ [Δ(4, 3) : L(μ)] \\ [Δ(4, 2, 1) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix})

We determine now the decomposition matrix for the principal block of the Hecke algebra. By using similar arguments given for l = 3 and p ≠ 2 we see that this is of the form

(\begin{matrix}[S^{(7)} : D^{μ}] \\ [S^{(5,2)} : D^{μ}] \\ [S^{(4,3)} : D^{μ}] \\ [S^{(4,2,1)} : D^{μ}] \end{matrix}) = (\begin{matrix}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{matrix})

We return to the q-Schur algebra. The projective P (4, 2, 1) is a tilting module (isomorphic to T (5, 2)), and hence is also injective, its radical layers are

L(4, 2, 1)
L(4, 3)
L(5, 2) ⊕ L(4, 2, 1)
L(4, 3)
L(4, 2, 1)

This gives the structure of Δ(5, 2). The tilting module T (7) has quotients Δ(7), Δ(5, 2) and Δ(4, 2, 1). It cannot be P (4, 2, 1), so it does not have a simple top. Hence the top and socle must be a direct sum L(4, 2, 1) ⊕ L(5, 2) and we deduce that T (7) is ′D̃′_{6}-shaped. The reason why this is important is that it shows that the (unique) non-split extension of Δ(5, 2) and Δ(7) has a “tree shape” and is isomorphic to P (5, 2). The radical layering of P (5, 2) is given by

L(5, 2)
L(4, 3)
L(7) ⊕L(4, 2, 1)
L(4, 3)
L(5, 2)

That is, L(4, 2, 1) is in the socle and the factor module is uniserial. As for the structure of P (4, 3), its radical is a direct sum of Δ(7) with the second radical of P (4, 2, 1), and there is an exact sequence

0 → Δ(7) → P (4, 3) → rad_{S_{q}(3,7)}P (4, 2, 1) → 0

The structure of this algebra is similar to that of the algebra in [DEMN, 3.6]. It is wild, by a similar argument (which is actually not quite correct in [DEMN, 3.6], the zero relation was ignored but we will prove it here). Consider the quiver given in Figure 7 where the

only relation is αβ = 0. This is wild, it contracts to C D^{≈}_{4} (see [Ri1, p.168]).

It is interesting to note that although the decomposition matrices for the 2-part partitions of H_{q}(7) are the same for p = 2 and for p ≠ 2, the Young modules are different. If p ≠ 2 then Y^{(4,2)} has two Specht quotients but if p = 2 it has three.

[figure: L(4, 2, 1); L(4, 3); L(7); L(4, 3); L(5, 2); L(5, 2); L(4, 2, 1); L(5, 2); β; α] Figure 7

Next consider the algebra S_{q}(3, 8) for l = 3 and p = 2. Let us focus our attention on the block containing L(7, 1). The decomposition matrix for the Weyl modules is

(\begin{matrix}[Δ(7, 1) : L(μ)] \\ [Δ(6, 2) : L(μ)] \\ [Δ(4, 4) : L(μ)] \\ [Δ(4, 2^{2}) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix})

We determine as for S_{q}(3, 7) (the previous case) the decomposition numbers for the block of the Hecke algebra and use this to determine tilting modules for the q-Schur algebra. The algebra we obtain is Morita equivalent to the principal block component of S_{q}(3, 7) when l = 3, and p = 2. Therefore, S_{q}(3, 8) is of wild representation type.

5.4 We consider the algebra S_{q}(3, 4) for l = 2. We will show that it is Morita equivalent to R_{4}, and this is tame (see the proof of [DEMN, 5.7]). By using methods as before one shows easily that decomposition matrices are

(\begin{matrix}[Δ(4) : L(μ)] \\ [Δ(3, 1) : L(μ)] \\ [Δ(2^{2}) : L(μ)] \\ [Δ(2, 1^{2}) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix}) , (\begin{matrix}[S^{(4)} : D^{μ}] \\ [S^{(3,1)} : D^{μ}] \\ [S^{(2,2)} : D^{μ}] \\ [S^{(2,1^{2})} : D^{μ}] \end{matrix}) = (\begin{matrix}1 & 0 \\ 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix})

Using this and §2.6 we see that Y^{(3,1)} is uniserial with socle and top k with heart D^{(3,1)}. The module Y^{(2^{2})} is also uniserial of length three with socle and top D^{(3,1)} with heart k. Moreover Y^{(2,1^{2})} is the projective cover of D^{(3,1)} which is uniserial of length five with composition factors D^{(3,1)}, k, D^{(3,1)}, k, D^{(3,1)}. (Here we use that Y^{(2,2)} is a quotient of P (3, 1)).

The algebra End_{H_{q}(4)}(⊕_{λ∈Λ}Y^{λ}) is the basic algebra for S_{q}(3, 4). The Young modules satisfy (essentially) the hypotheses in [Erd2, 3.12] (the composition factor S_{3} in [Erd2, 3.12] is not visible in the endomorphism ring), and hence the basic algebra is isomorphic to the algebra R_{4}.

5.5 Consider now the algebra S_{q}(3, 5) for l = 2. We will show that the principal block component is Morita equivalent to H_{4} of [Erd2, 3.13], which is tame. The other block is semi-simple and therefore we will have shown that S_{q}(3, 5) is tame.

The decomposition matrices for the principal block components are as follows:

(\begin{matrix}[Δ(5) : L(μ)] \\ [Δ(3, 2) : L(μ)] \\ [Δ(3, 1^{2}) : L(μ)] \\ [Δ(2^{2}, 1) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix}) , (\begin{matrix}[S^{(5)} : D^{μ}] \\ [S^{(3,2)} : D^{μ}] \\ [S^{(3,1^{2})} : D^{μ}] \\ [S^{(2^{2},1)} : D^{μ}] \end{matrix}) = (\begin{matrix}1 & 0 \\ 0 & 1 \\ 1 & 1 \\ 0 & 1 \end{matrix})

To prove this, we find first the decomposition matrix for the Weyl modules using techniques that have already been demonstrated in previous examples. ¿From this we see that the Young module Y^{(3,2)} is just S^{(3,2)} and hence S^{(3,2)} is simple. Then the rest of the decomposition matrix for the Hecke algebra follows easily.

The Young module Y^{(3,1^{2})} has length four and socle and top each are isomorphic to k ⊕ D^{(3,2)}. Moreover Y^{(2^{2},1)} is the projective cover of D^{(3,2)} and it has Loewy length three, the middle is a direct sum k⊕D^{(3,2)}. The endomorphism ring of ⊕_{λ}Y^{λ} is the basic algebra of the principal block of S_{q}(3, 5). This algebra is isomorphic to H_{4}, as in [Erd2, 3.13]. Actually, this is proved using the arguments of [Erd2, 3.14] by letting E_{1} = k, E_{2} = D^{(3,2)} and E_{3} = Y^{(3,1^{2})} and E_{4} = P (D^{(3,2)}). The crucial ingredient is that there is a homomorphism π : E_{4} → E_{3} and a homomorphism j : E_{3} → E_{4} such that the composition jπ has image D^{(3,2)} ⊕ k. This gives the commutativity relation.

5.6 We will now prove that the algebra S_{q}(3, 6) for l = 2 is wild. The decomposition matrix for the Weyl modules is given as follows:

(\begin{matrix}[Δ(6) : L(μ)] \\ [Δ(5, 1) : L(μ)] \\ [Δ(4, 2) : L(μ)] \\ [Δ(4, 1^{2}) : L(μ)] \\ [Δ(3^{2}) : L(μ)] \\ [Δ(2^{3}) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 1 & 0(1) & 0 & 0 & 0 \\ 0 & 1 & 1 & ∗ & 0 & ∗ \\ 0 & 0 & 1 & 1 & 1 & 1(2) \\ 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix})

Here the composition factors in brackets occur for p = 3 only. We proceed now by using the argument as in Proposition 5.2(A), for the algebra S_{q}(3, 10) when l = 3. We see from the decomposition matrix that the tilting modules T (3^{2}) and T (4, 1^{2}) are uniserial of length three, with top and socle L(2^{3}) and the heart L(3^{2}) and L(4, 1^{2}) respectively. So there is an exact sequence

0 → W → P (2^{3}) → X → 0

and an exact sequence

0 → Δ(3^{2}) ⊕ Δ(4, 1^{2}) → X → L(2^{3}) → 0

where W has quotient Δ(4, 2). By the same argument used in Proposition 5.2(A) it follows that the algebra S_{q}(3, 6) for l = 2 is wild.

5.7 Next we show that the algebra S_{q}(3, 7) for l = 2 is wild. The line of reasoning will be similar to the one used to show that S_{q}(3, 11) is wild when l = 3 and p ≠ 2. The decomposition matrix of the principal block component is of the form

(\begin{matrix}[Δ(7) : L(μ)] \\ [Δ(5, 2) : L(μ)] \\ [Δ(5, 1^{2}) : L(μ)] \\ [Δ(4, 2, 1) : L(μ)] \\ [Δ(3^{2}, 1) : L(μ)] \\ [Δ(3, 2^{2}) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 0(1) & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & ∗ & ∗ & ∗ \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix})

The extra composition factor for Δ(7) exists only when p = 3. The submatrix corresponding to the 3-part partitions is the same as for S_{q}(3, 4) and actually the corresponding quotient of S_{q}(3, 7) is Morita equivalent to S_{q}(3, 4), an equivalence is given by tensoring with the q-determinant, hence is Morita equivalent to R_{4}.

Assume first that p ≠ 3, then there is a non-split extension of L(7) with L(5, 1^{2}). Combining this with the information on 3-part partitions we see that the quiver contains

L(7) ⟷ L(5, 1^{2}) ⟷ L(4, 2, 1) ⟷ L(3^{2}, 1) ⟷ L(3, 2^{2}).

There must be arrows joining this quiver with L(5, 2) and by the argument as for S_{q}(3, 11) and l = 3, p ≠ 2 (Proposition 5.2(B)) we get a wild algebra.

Now assume p = 3, then the Weyl module Δ(7) is uniserial, with heart L(5, 2) and socle L(5, 1^{2}), so we get a subquiver

(7) ⟷ (5, 2) ⟷ (5, 1^{2}) ⟷ (4, 2, 1) ⟷ (3^{2}, 1) ⟷ (3, 2^{2})

Suppose, for a contradiction, this is the full quiver. Then we use the argument as in the proof of 4.3(B) and deduce that the radical of Δ(5, 2) must be uniserial, with composition factors L(5, 1^{2}), L(4, 2, 1), L(3^{2}, 1), L(3, 2^{2}), so there is an epimorphism π : P (5, 1^{2}) → rad(Δ(5, 2)) which must factor through Δ(5, 1^{2}), a contradiction.

5.8 Finally, we will demonstrate that the algebra S_{q}(3, 8) for l = 2 is wild. If p = 3 this follows already from §3.4, so we assume p ≠ 3. A part of the decomposition matrix given as follows.

(\begin{matrix}[Δ(8) : L(μ)] \\ [Δ(7, 1) : L(μ)] \\ [Δ(6, 2) : L(μ)] \\ [Δ(5, 3) : L(μ)] \\ [Δ(4^{2}) : L(μ)] \\ [Δ(6, 1^{2}) : L(μ)] \\ [Δ(4, 3, 1) : L(μ)] \\ [Δ(4, 2^{2}) : L(μ)] \\ [Δ(3^{2}, 2) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & ∗ & ∗ & ∗ & ∗ \\ 0 & 0 & 0 & 1 & 1 & 0 & ∗ & ∗ & ∗ \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix})

The quotient algebra corresponding to partitions with exactly three parts is Morita equivalent to S_{q}(3, 5), that is to the algebra H_{4}. This gives also part of the quiver. The part corresponding to 2-part partitions is obtained by row-and column removal.

We wish to complete the row for Δ(7, 1), that is the last four entries. By the dimensions, these follow if we show that L(6, 1^{2}) must occur. Now L(5, 1^{2}) occurs in Δ(7), and hence L(6, 1^{2}) ≅ L(5, 1^{2}) ⊗ L(1) occurs in Δ(7) ⊗ L(1), which has Δ-quotients with weights (8) and (7, 1), and the claim follows.

Now we consider the quiver, it contains L(8) ⟷ L(7, 1), and a subquiver of type H_{4} (Figure 5) with X_{1} = L(6, 1^{2}), X_{2} = L(4, 2^{2}), X_{3} = L(3^{2}, 2) and X_{4} = L(4, 3, 1). Moreover, from the decomposition matrix we see that there are arrows L(4^{2}) ⟷ L(4, 3, 1). So we have a subquiver of shape A_{2} and a subquiver of shape D_{5}. Next consider how L(6, 2) is joined. There are three possibilities for the submodule structure of Δ(7, 1), two of them

lead to a subquiver which contracts to D^{≈}_{4} and gives a wild algebra. So we are left with the possibility that Δ(7, 1) is uniserial with second socle L(6, 2) in which case the shape of the quiver got so far is E_{8}. The vertex corresponding to L(5, 3) must be joined to at least one of L(4^{2}) or L(4, 3, 1) since there is a projective with Δ-quotients Δ(4^{2}) and Δ(5, 3) which has

a simple top. In both cases the quiver leads to a wild algebra (either E^{≈}_{6} or after contraction

to D^{≈}_{4}).

6 Representation type for some S_{q}(4, r) and S_{q}(5, r)

6.1 We first show that S_{q}(4, 7) for p ≠ 2 and l = 3 is wild. The Weyl module Δ(7) is uniserial of length three, with heart L(5, 2) and socle L(2^{3}, 1). Hence, the quiver contains

L(7) ⟷ L(5, 2) ⟷ L(2^{3}, 1).

The structure of S_{q}(4, 3)-modules imply that the quiver of S_{q}(4, 7) also contains

L(4, 1^{3}) ⟷ L(3, 2, 1^{2}) ⟷ L(2^{3}, 1).

Consider the principal block of S_{q}(4, 4) for the same field. There exists a non-split exact sequence

0 → L(1^{4}) → Δ(2^{2}) → L(2, 2) → 0.  (6.1.1)

Tensoring this exact sequence with L(1)^{F} gives a non-split exact sequence

0 → L(4, 1^{3}) → Δ(2^{2}) ⊗ L(1)^{F} → L(5, 2) → 0.  (6.1.2)

Therefore, the quiver also contains L(4, 1^{3}) ⟷ L(5, 2). Combining this information we obtain a subquiver which is an “extended circuit”, thus by Proposition 2.3(B), S_{q}(4, 7) for p ≠ 2 and l = 3 is wild.

6.2 Now we will show that the algebra S_{q}(4, 8) for p ≠ 2 and l = 3 is wild. The Weyl module Δ(8) is uniserial of length three with socle L(2^{4}) and heart L(5, 2, 1). By tensoring Weyl modules for S_{q}(4, 4) with the determinant we see that Δ(5, 1^{3}) has length two and socle L(3^{2}, 1^{2}), and Δ(3^{2}, 1^{2}) has also length two and socle L(2^{4}).

Consider S_{q}(4, 5) over the same field. There is a non-split exact sequence

0 → L(2, 1^{3}) → N → L(2^{2}, 1) → 0.  (6.2.1)

Tensoring this sequence with L(1)^{F} gives a non-split exact sequence with end terms L(5, 1^{3}) and L(5, 2, 1). Combining this information yields a “circuit” of length four which is joined to another vertex. Hence, by Proposition 2.3(B) S_{q}(4, 8) is wild when p ≠ 2 and l = 3.

6.3 Consider the algebra S_{q}(4, 4) for l = 2. Here the Weyl module Δ(4) is uniserial of length three with socle L(1^{4}) and heart L(3, 1). The decomposition matrix for Weyl modules is given by

(\begin{matrix}[Δ(4) : L(μ)] \\ [Δ(3, 1) : L(μ)] \\ [Δ(2^{2}) : L(μ) \\ [Δ(2, 1^{2}) : L(μ)] \\ [Δ(1^{4}) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

The upper left 4 × 4 part is obtained from S_{q}(4, 4). For the last column, note that P (1^{4}) is a tilting module and it must be T (4). By considering the decomposition matrix for the Hecke algebra H_{q}(4) (see 5.4, add a row (1, 0) for S^{(1^{4})}) we get its Δ- quotients.

From the decomposition matrix and the structure of Δ(4) one sees the quiver contains

L(4) ⟷ L(3, 1) ⟷ L(1^{4}) ⟷ L(2, 1^{2}) ⟷ L(2^{2}).

The projective module P (2^{2}) has a filtration with quotients Δ(2^{2}) and Δ(3, 1) (by reciprocity). It has a simple top, hence there must be a non-split extension of L(3, 1) with either L(2^{2}) or with L(2, 1^{2}). In both cases the quiver contains a proper subquiver which is a circuit and by Proposition 2.3(B) the algebra is wild.

6.4 Let l = 2 and consider S_{q}(4, 5). First, the principal block of S_{q}(4, 5) has only partitions with three parts, so it is Morita equivalent to the principal block of S_{q}(3, 5) and is therefore tame. The other block has only two partitions and is of finite type. Hence, S_{q}(4, 5) is of tame representation type for l = 2.

6.5 Finally, consider the algebra S_{q}(5, 5) for l = 2. We will show that the prinicpal block is wild. First we claim that the decomposition matrix for Weyl modules is as follows:

(\begin{matrix}[Δ(5) : L(μ)] \\ [Δ(3, 2) : L(μ)] \\ [Δ(3, 1^{2}) : L(μ)] \\ [Δ(2^{2}, 1) : L(μ)] \\ [Δ(1^{5}) : L(μ)] \end{matrix}) = (\begin{matrix}1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

The upper left 4×4 part of the decomposition matrix of the Weyl modules is obtained from S_{q}(3, 5). It remains to determine the three entries in the last column. Note that P (1^{5}) is a tilting module and it must be T (5). The last column is now obtained by the argument of 6.3 (using 5.5).

The Weyl module Δ(5) is uniserial, with middle L(3, 1^{2}) and socle L(1^{5}). Next observe that P (2^{2}, 1) is a tilting module and hence is T (3, 2). So its socle is simple L(2^{2}, 1) and this must be the socle of Δ(3, 2). We deduce that Δ(3, 2) is uniserial, with heart L(3, 1^{2}).

Combining these we see that the quiver contains a D̃_{4}-shaped subquiver with center L(3, 1^{2}). Since the quiver must be visible by looking at the second radical layer of Weyl modules, we deduce that Δ(3, 1^{2}) must have semisimple radical, L(2^{2}, 1) ⊕ L(1^{5}). Now we obtain as the radical layering of P (2^{2}, 1) ≅ I(2^{2}, 1) = T (3, 2)

L(2^{2}, 1)
L(3, 1^{2})
L(3, 2) ⊕ L(2^{2}, 1) ⊕ L(1^{5})
L(3, 1^{2})
L(2^{2}, 1)

This shows that the algebra S_{q}(5, 5) for l = 2 is wild by comparison with the representations

of a quiver of shape C D^{≈}_{4}.

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Mathematical Institute, Oxford University, 24-29 St. Giles, Oxford, OX1 3LB

E-mail address: erdmann@maths.ox.ac.uk

Department of Mathematics and Statistics, Utah St. University, Logan, UT. 84322, U.S.A.

E-mail address: nakano@sunfs.math.usu.edu